Quasi-periodic dynamics and a Neimark-Sacker bifurcation in nonlinear random walks on complex networks

We study the dynamics of nonlinear random walks on complex networks. We investigate the role and effect of directed network topologies on long-term dynamics. While a period-doubling bifurcation to alternating patterns occurs at a critical bias parameter value, we find that some directed structures give rise to a different kind of bifurcation that gives rise to quasi-periodic dynamics. This does not occur for all directed network structure, but only when the network structure is sufficiently directed. We find that the onset of quasi-periodic dynamics is the result of a Neimark-Sacker bifurcation, where a pair of complex-conjugate eigenvalues of the system Jacobian passes through the unit circle, destabilizing the stationary distribution with high-dimensional rotations. We investigate the nature of these bifurcations, study the onset of quasi-periodic dynamics as network structure is tuned to be more directed, and present an analytically tractable case of a four-neighbor ring.